Signals and systems convolution pdf

The general formula for correlation is. It is defined as correlation of a signal with itself. Consider a signals x t. The auto correlation function of x t with its time delayed version is given by. Auto correlation function of energy signal at origin i.

Consider two signals x 1 t and x 2 t. Cross correlation function corresponds to the multiplication of spectrums of one signal to the complex conjugate of spectrum of another signal.

Parseval's theorem for energy signals states that the total energy in a signal can be obtained by the spectrum of the signal as. Gowthami Swarna. Convolution and Correlation Advertisements. Previous Page. Next Page. Useful Video Courses.

Time reflection Time reflection is given by equation 1. Time shifting The equation representing time shifting is given by equation 1. Time shifting and scaling The combined transformation of shifting and scaling is contained in equation 1. Here, time shift has a higher precedence than time scale. The signal has to be shifted first and then time scaled.

Exponential signals: The exponential signal given by equation 1. Let us consider a decaying exponential signal. The magnitude of this signal at five times the time constant is,. Hence, in most engineering applications, the exponential signal can be said to have reached its final value in about ten times the time constant. If the time constant is 1 second, then final value is achieved in 10 seconds!! We have some examples of the exponential signal in figure 1. Fig 1. The sinusoidal signal: The sinusoidal continuous time periodic signal is given by equation 1.

We have the complex exponential signal given by equation 1. From equation 1. The sketch of this is given in fig 1. This is a function which cannot be practically generated. Figure 1. The unit step The unit step function, usually represented as u t , is given by,. The unit ramp: The unit ramp function, usually represented as r t , is given by,. The signum function: The signum function, usually represented as sgn t , is given by.

This article basically deals with system connected in series or parallel. This condition, denoted by BIBO, can be represented by:. Some examples of stable and unstable systems are given in figure 1. Memory The system is memory-less if its instantaneous output depends only on the current input.

In memory-less systems, the output does not depend on the previous or the future input. Examples of memory less systems:. Causality: A system is causal, if its output at any instant depends on the current and past values of input.

The output of a causal system does not depend on the future values of input. This can be represented as:. For a causal system, the output should occur only after the input is applied, hence, x[n] 0 for n 0 implies y[n] 0 for n 0 All physical systems are causal examples in figure 7.

Non-causal systems do not exist. This classification of a system may seem redundant. But, it is not so. This is because, sometimes, it may be necessary to design systems for given specifications.

When a system design problem is attempted, it becomes necessary to test the causality of the system, which if not satisfied, cannot be realized by any means.

Hypothetical examples of non-causal systems are given in figure below. Linearity: The system is a device which accepts a signal, transforms it to another desirable signal, and is available at its output. We give the signal to the system, because the output is s. Time invariance: A system is time invariant, if its output depends on the input applied, and not on the time of application of the input. Hence, time invariant systems, give delayed outputs for delayed inputs.

Throughout the rest of the course we shall be dealing with LTI systems. If the output of the system is known for a particular input, it is possible to obtain the output for a number of other inputs. We shall see through examples, the procedure to compute the output from a given input-output relation, for LTI systems.

Example — I:. Figure 1: The continuous time system Some of the different methods of representing the continuous time system are: i Differential equation ii Block diagram iii Impulse response iv Frequency response v Laplace-transform vi Pole-zero plot. Moreover, from each of the above representations, it is possible to obtain the system properties using parameters as: stability, causality, linearity, invertibility etc.

We now attempt to develop the convolution integral. Usually the impulse response is denoted by h t. Methods of evaluating the convolution sum: Given the system impulse response h[n], and the input x[n], the system output y[n], is given by the convolution sum:.

Problem: To obtain the digital system output y[n], given the system impulse response h[n], and the system input x[n] as:. Evaluation as the weighted sum of individual responses The convolution sum of equation … , can be equivalently represented as: y[n] Evaluation using graphical representation: Another method of computing the convolution is through the direct computation of each value of the output y[n].

This method is based on evaluation of the convolution sum for a single value of n, and varying n over all possible values. Example: A system has impulse response h[n] exp 0.

Obtain the unit step response. Evaluation from Z-transforms: Another method of computing the convolution of two sequences is through use of Z-transforms. This method will be discussed later while doing Z-transforms. This approach converts convolution to multiplication in the transformed domain. Evaluation from Discrete Time Fourier transform DTFT : It is possible to compute the convolution of two sequences by transforming them to the frequency domain through application of the Discrete Fourier Transform.

This approach also converts the convolution operator to multiplication. Since efficient algorithms for DFT computation exist, this method is often used during software implementation of the convolution operator.

Methods of evaluating the convolution integral: Same as Convolution sum Given the system impulse response h t , and the input x t , the system output y t , is given by the convolution integral. Some of the different methods of evaluating the convolution integral are: Graphical representation, Mathematical equation, Laplace-transforms, Fourier Transform, Differential equation, Block diagram representation, and finally by going to the digital domain.

For any system, we can define its impulse response as: h t y t when x t t For linear time invariant system, the output can be modeled as the convolution of the impulse response of the system with the input. Let x t be the input voltage source and y t be the output current. Then summing up the voltage drops around the loop gives. Solving differential equation: A wide variety of continuous time systems are described the linear differential equations:.

Fourier series has long provided one of the principal methods of analysis for mathematical physics, engineering, and signal processing. It has spurred generalizations and applications that continue to develop right up to the present. While the original theory of Fourier series applies to periodic functions occurring in wave motion, such as with light and sound, its generalizations often relate to wider settings, such as the time-frequency analysis underlying the recent theories of wavelet analysis and local trigonometric analysis.

These arguments were still imprecise and it remained for P. Dirichlet in to provide precise conditions under which a periodic signal could be represented by a FS. We have seen in previous chapters how advantageous it is in LTI systems to represent signals as a linear combinations of basic signals having the following properties.

To represent signals as linear combinations of basic signals. Set of basic signals used to construct a broad class of signals. The response of an LTI system to each signal should be simple enough in structure. It then provides us with a convenient representation for the response of the system. Response is then a linear combination of basic signal. Historical background There are antecedents to the notion of Fourier series in the work of Euler and D.

Bernoulli on vibrating strings, but the theory of Fourier series truly began with the profound work of Fourier on heat conduction at the beginning of the century.

In [5], Fourier deals with the problem of describing the evolution of the temperature of a thin wire of length X. He proposed that the initial temperature could be expanded in a series of sine functions:. The following relationships can be readily established, and will be used in subsequent sections for derivation of useful formulas for the unknown Fourier coefficients, in both time and frequency domains.

T cos kw0t cos gw0t dt 0 5 0. Also, k and g are integers. T 1 B sin 2kw0T 0 14 2 4kw0. From Equation 1 , T [sin g k w0 t ]dt 0 0.

Adding Equations 15 , 19 , T T 2C sin gw0 t cos kw0 t dt sin kw0 t cos gw0 t dt 0 0. Prove that T sin kw0 t sin gw0 t dt 0 0. Since cos cos cos sin sin or sin sin cos cos cos Thus, T T D cos kw0 t cos gw0 t dt cos k g w0 t dt 23 0 0.

Adding Equations 23 , 26 T T 2D sin kw0 t sin gw0 t cos kw0 t cos gw0 t dt 0 0. Properties of Fourier Representation 1. Linear Properties 2. Translation or Time Shift Properties 3. Frequency shift properties 4. Time Differentiation 5. Time Domain Convolution 6. Modulation or Multiplication Theorem 7.

Frequency shift properties. The z-transform is a transform for sequences. Just like the Laplace transform takes a function of t and replaces it with another function of an auxiliary variable s. The z-transform takes a sequence and replaces it with a function of an auxiliary variable, z.

The reason for doing this is that it makes difference equations easier to solve, again, this is very like what happens with the Laplace transform, where taking the Laplace transform makes it easier to solve differential equations. Difference equations arise in numerical treatments of differential equations, in discrete time sampling and when studying systems that are intrinsically discrete, such as population models in ecology and epidemiology and mathematical modelling of mylinated nerves.

Generalizes the complex sinusoidal representations of DTFT to more generalized representation using complex exponential signals.

You can see that when you do the z-transformit sums up all the sequence, and so the individual terms affect the dependence on z, but the resulting function is just a function of z, it has no k in it. It will become clearer later why we might do this. Partial fraction method 2. Power series method 3. Long division method Partial fraction method:.

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